DIGITAL signal processing... or just Signal Processing?

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1 ELECTRICAL AND SYSTEMS ENGINEERING DRAFT DECEMBER 3, ESE 482. Digital Signal Processing R. Martin Arthur Abstract Introduction to analysis and synthesis of discretetime linear shift-invariant (LSI) systems. Discrete-time convolution, discrete-time Fourier transform, z-transform, rational function descriptions of discrete-time LSI systems. Sampling, analog-to-digital conversion, and digital processing of analog signals. Techniques for the design of finite impulse response (FIR) and infinite impulse response (IIR) digital filters. Hardware implementation of digital filters and finite-register effects. The Discrete Fourier Transform and the Fast Fourier Transform (FFT) algorithm. Prerequisite: ESE 351. I. Lecture #1 TAKE THE ROLL on WEDNESDAY A. Introduction DIGITAL signal processing... or just Signal Processing? cf., food processing 1) Objective: To develop quantitative methods to sample, characterize, compress, analyze, interpolate, and recover signals and to design discrete-time systems. That objective does not mean we abandon our judgment and experience in evaluating and interpreting the results of those methods. 2) Prerequisite: Introduction to signals and systems. Topics include Characterization of signals. Example std08n.pdf Solution of differential equations B. Website Show intro-351.a7.ppt in /demo Show algebra.ppt in /demo 1) Catalog description (changes). 2) Signals: two-tone suppression, maury-cheveau maneuver, us thermometry, rma ecg work with mpg 3) rma website 4) Syllabus 5) Handouts: key expressions, etc. 6) Homework will be posted here. Be sure to check for assignments and DUE dates. 7) Office hours, test, final 8) Ethics 9) Matlab Run demosine.m in ese482/mfls R. M. Arthur is with the Department of Electrical and Systems Engineering, Washington University School of Engineering. rma@ese.wustl.edu C. Signal Characteristics Signals: Continuous x(t) and discrete x[n] Run rampstairs.m in ese482/mfls Sample (S/H) Continuous to Discrete Interpolate, Low-Pass Filter Discrete to Continuous Representation in a computer Quantize (A/D) Discrete to Digital Convert (D/A) Digital to Discrete Time Signal-to-noise ratio (SNR): We need to know What s signal? What s noise? Run rnsmplsfn.m in ese482/mfls Source of the signals, noise Gaussian and Poisson (PET, Google) signals? D. Prerequisite Quiz Go over solution to the prerequisite quiz A. Prerequisite Examples 1) Convolution: II. LECTURE #2 System Output Fast convolution using the Fourier transform 2) Transforms: Frequency content of signals Run pltrmadipx.m in ese482/mfls Solution of differential equations RLC circuit with sinusoidal excitation (source) Solve using matrix algebra & the Fourier transform Run lpfcheb.asc in ese482/mfls B. Input/Output Model Input System Output In general there may be many levels of abstraction for a problem. Specifically, in our context there may be many subsystems, i.e., multiple transfer functions that apply to the same problem (state space?). What other signal characteristics would you like to add? Bounded signal? Stable system? What s the difference? NONE for linear signals and systems!

2 2 ELECTRICAL AND SYSTEMS ENGINEERING DRAFT DECEMBER 3, ) Specific Signals: Sinusoids? Exponentials? e jωt e jωn e jωt = cos(ωt) + j sin(ωt) Impulse? δ(t) Unit-Sample? δ[n] 1) Signal Processing: Definition of a Signal Domain & Range 2) Definition of a sequence. Examples: Unit-sample sequence Unit-step sequence 3) Basic definitions Linearity Shift Invariance Unit-Sample Response 4) Examples: Run in /mfls thismom.m demosine.m demontrs.m demontrr.m C. Homework #1 Go over Homework assignment * Go over Quiz on Homework #1 on 3 September The form of the solutions and Matlab figure labeling and program documentation. PREPARE DETAILED, SELF-CONTAINED SOLUTIONS, NOT ANSWERS! (The answers only certify the process & are worth <50%.) Solution to Homework Problem 1.10 as example A digital communication link carries binary-coded words representing samples of an input signal x a (t) = 3 cos 600πt + 2 cos 1800πt (1) The link is operated at 10,000 bits/s and each input sample is quantized into 1024 different voltage levels a) What are the sampling and folding frequencies? b) What is the Nyquist rate for the signal x a (t)? c) What are the frequencies in the resulting discrete-time signal x[n]? d) What is the resolution? SOLUTION Problem Statement: Given a 10kbit link carrying 10-bit samples, find the sample frequency F s, the folding frequency, and the Nyquist rate for x a (t). Find the frequencies in x[n]. Find the distance between quantization levels. Basic definitions Convolution Properties Stability Causality III. LECTURE #3 Sampling: Let x(t) = A cos(ωt) for t = nt, then ω = ΩT Run samsine.m in /mfls Solve homework problems Problem 8. PS: Find the Nyq. Rate for 100 Hz signal. Find the folding frequency for a sample rate of 250s/s Problem 11. PS: Find the output of a discrete-time system that is a 5:1 interpolator of an aliased signal obtained by sampling x a (t) = 3 cos 100πt + 2 sin 250πt at 200 s/s Problem 12. worked in Example Problem 13. PS: Find the number of bits of quantization required to obtain 0.1 and 0.02 resolutions for a ± 6.35 amplitude signal. Administer Quiz 1 IV. LECTURE #4 Go over Quiz #1 Solution and scores Run q1b4.m in /qz1 Comment on Homework #2 and the Answers LTI and LSI system descriptions N th order constant-coefficient differential equations N th order constant-coefficient difference equations Fully documented Matlab scripts and functions Go over contents of demosine.m in /mfls and on website V. LECTURE #5 Go over Homework assignment #3 demosine.m The form of the solutions Matlab problem with fully documented mfile Matlab figure labeling and program documentation Demonstrate genexpt.m in /mfls Run demodeqn.m in /mfls Solution of DEs 1) Direct method: Find the homogeneous and particular solutions y[n] = y h [n] + y p [n] (2) 2) Homogenous solution is always an exponential: y h [n] = λ n (3) Substitute y h [n] into the DE to find the roots of the characteristic equation y h [n] = C 1 λ n 1 + C 2 λ n 2 + C 3 λ n (4) 3) The particular solution y p [n] has the form of the forcing function 4) Combine y h [n] and y p [n] to form the complete solution 5) Use the initial conditions to find the C i s Example solving difference equations Problem 2.49 solving difference equations: build h[n] for S from unit-sample response alone by superposition Administer Quiz 2

3 ARTHUR: ESE 482. DIGITAL SIGNAL PROCESSING DECEMBER 3, VI. LECTURE #6 Quiz #2 Scores on website Run qz2482fb4.m in /quizzes/qz2 Questions on Homework #3? the Matlab exercise? Revisit problem Solve the difference equation, which describes S, by building h[n] from the unit-sample response alone by applying linearity, i.e., using superposition, proportionality and shift invariance. Given y[n] 0.8y[n 1] = 2x[n] + 3x[n 1], then y h [n] = c(0.8) n. Consider simpler system S 1 y[n] 0.8y[n 1] = x 1 [n] = δ[n]. Because y[0] = 1, c = 1 for y 1 [n] = (0.8) n u[n] Consider simpler system S 2 and use shift invariance, y[n] 0.8y[n 1] = x 2 [n 1] = δ[n 1]. Because y[1] = 1, c = 1 for y 2 [n] = (0.8) n 1 u[n 1]. Scale and add y 1 [n] and y 2 [n] to get h[n]. Z transform definition, sampling the Laplace transform X(z) = x[n]z n, for z = e st (5) Inverse Z transforms Power series Long division Partial-fraction expansion Residue theorem VII. LECTURE #7 Review Matlab exercise requirements Go over Homework assignment #4 Z transform properties. See Table 3.2 Describe Figure in /lecture Run pmfig335.m in /mfls Questions on Homework #3 problems Inverse Z transforms Administer Quiz 3 VIII. LECTURE #8 Go over Quiz #3 Solution and scores Run qz3482fb4.m in /quizzes/qz3 Inverse Z transforms Power series Long division Partial-fraction expansion Residue theorem Run invac.m in /mfls IX. LECTURE #9 Go over Homework assignment #5 DUE 1 October Matlab exercise #2 requirements Direct forms I and II and their equivalence Relation of poles & zeros to the frequency response Causality and Stability One-sided z-transform; Solve Problem 3.49 part a Solve problem 3.42: Find h[n] and ZS s[n] z z 2 H(z) = 1 0.6z z 2 (6) Run pmprb342.m in /mfls Run pmprb345.m in /mfls Run pmprb349.m with IC: y[-2]=1,y[-1]=1 in /mfls Administer Quiz 4 X. LECTURE #10 Go over Quiz #4 Solution and scores Frequency Analysis Run demodtft.m in /mfls to demonstrate zero-phase Run fosysfr.m in /mfls to connect genexpt.m exercise result with first-order system behavior Run pltrmadipx.m in /mfls to characterize an ECG for processing, diagnosis, and instrument design Frequency Response: Fourier Transform 1) Characterize a signal for frequency content instrument design equivalent signal representation 2) Perform convolution Factored H(z) on the unit circle H(e jω ): Frequency response Gain in db Frequency response Phase in radians XI. LECTURE #11 Frequency response of a first-order system Effect of a single pole on H(e jω ) Effect of a single zero on H(e jω ) Run iirfireq.m in /mfls to show equivalent FIR pole & zero plot for a first-order (single pole) IIR system Run demompgd.m in /mfls to show effect of poles on group delay Fourier Series & Transform - continuous & discrete Questions on Chapter 4 Administer Quiz 5

4 4 ELECTRICAL AND SYSTEMS ENGINEERING DRAFT DECEMBER 3, 2014 XVI. LECTURE #16 XII. LECTURE #12 Go over Quiz 5: Run qz5fb4.m in /qz5 Go over Matlab Exercise #2: Run ml2fa9.m in /ml2 1) Fourier Series & Transform - continuous & discrete 2) Frequency-Domain Analysis of LSI Systems 3) Effects of an LSI system on sinusoids XIII. SESSION #13 - TEST 1 Test on Wednesday 8 Oct: 4 problems. Each problem similar to a quiz problem Material discussed in class through Fourier transforms of continuous- & discrete-time systems Homework assignments #1 through #5 Assigned text material through chapter 4 on Fourier transforms for both continuous functions and discrete sequences Appropriate subset of tables from the website to be available during the test XIV. LECTURE #14 Go over Test Go over Homework #6 and Matlab Assignment Properties of the Fourier transform, H(e jω ) Find inverse Fourier transform of an ideal low-pass filter Linear-phase systems, example of H(e jω ) = e jωα. Questions on Homework #6 Go over Homework #7 Minimum Phase Transient and steady-state responses Run samsine.m in /mfls Sample Theorem Sampling Administer Quiz 6 XVII. LECTURE #17 Go over Quiz 6 Run qz6fb4.m in /quizzes/ Sampling Ideal Sampler & Practical Samplers Aliasing Sample & Hold Circuits XVIII. LECTURE #18 Group Delay: [ ] τ(ω) = dθ(ω) j dh(e jω ) d(ω) = Re dω H(e jω ) (7) Run demogdly.m in /mfls All Pass, Non-Minimum Phase, Minimum Phase systems Fig. 1. S/H, D/A, & A/D conversion from Proakis & Manolakis, 4th edition. Dispersive Model for Ultrasound Propagation, Ultrasonic Imaging, 4: , in /handouts Run demomnph.m in /mfls to demonstrate minimum phase systems XV. LECTURE #15 Group Delay in tests, circuits and systems Run lfpcheb.asc in /demo Run democheby1.m in /mfls [B,A]=CHEBY1(N,R,Wp), R pkpass ripple, Wp 0-1 Discrete-Time Signal Processing example Run differentiator.m in /demo System Design Chebyshev polynomials in chebypolys.pdf in /lecture Run chebypolys.m in /mfls Sampling & Reconstuction A/D Converters (See Figure??). D/A and A/D Converters Spectrum replication, p6.11 Run demop611.m in /mfls Linear interpolation system, p6.15 Sinc squared is triangle in time, see p a) h(t) a triangle Run pmprb615.m in /mfls H(f) = e j2πft T ( ) 2 sin(πft ) (8) πf H(f) = T e j2πft sinc(πft ) 2 (9) Scale factor, Delay, Sinc squared function Administer Quiz 7

5 ARTHUR: ESE 482. DIGITAL SIGNAL PROCESSING DECEMBER 3, XIX. LECTURE #19 Go over Quiz 7 Run qz7fb4.m in /quizzes/ Go over handout on key expressions Matlab exercise Go over Homework #8 Description & specifications in /homework Due Wednesday, December 3 Run bpfirpm.m Show fsapap91.pdf in /handouts XX. LECTURE #20 Convolution Linear Circular Discrete Fourier Series Discrete Fourier Transform Convolution with the DFT Run imptrain.m in /mlfs Run demofcon.m in /mfls z[111] 2 conv Run convsfcon.m in /mfls XXI. LECTURE #21 Revisit the exercise description Run demorlft.m in /mfls Run demosysd.m in /mfls Run demozpbt.m in /mfls Convolution with the DFT Linear Block Questions on Homework #8 Problems Administer Quiz 8 XXII. LECTURE #22 Go over Quiz 8 Run qz8fb4.m in /quiz8 Go over project description demobcel.m in /mfls Time Frequency Fast Fourier Transform XXIV. LECTURE #24 Go over the last homework assignment (#10), due 12/1 1) Number representation 2) Quantization A/D, coefficients Arithmetic operations Attenuator Questions on Homework #9 Problems Administer Quiz 9 XXV. LECTURE #25 Go over Quiz 9 Run qz9fb4.m in quizzes/qz9 Problem 9.36 (b) lhe nrsr nv1;; output" u... - v ~. ~ The digital system shown in Fig. P9.36 uses a six-bit (including sign) fixed-point twos-complement AID converter with rounding, and the filter H (z) is implemented using eight-bit (including sign) fixed-point twos-complement fractional arithmetic with rounding. The input x(t) is a zero-mean uniformly distribl:lted random process having autocorrelation YxxCr) = 38(r). Assume that the AID converter can handle input values up to ±1.0 without overflow. (a) What value of attenuation should be applied prior to the AID converter to assure that it does not overflow? (b) With the attenuation above, what is the signal-to-quantization-noise ratio (SQNR) at the AID converter output? _, (c) The six-bit AID samples can be left justified, right justified, or centered in the eight-bit word used as the input to the digital filter. What is the correct strategy to use for maximum SNR at the filter output without overflow? (d) What is the SNR at the output of the filter due to all quantization noise sources? Figure P9.36 Fig. 2. r Attenuator l,.,.. J AID converter 6 bits j Filter H(z) (8 bits) x(n) i---~--i + r y(n) 0.75 ~ ~ Problem 9.36 from Proakis & Manolakis, 4th edition. Quantization: Error propogation IIR first-order dead zone FIR structures DFT algorithm IIR structures XXIII. LECTURE #23 Questions on the Matlab Exercise Administer Bidirection QUIZ!!! FFT for N=2 ν = 8 for ν = 3

6 6 ELECTRICAL AND SYSTEMS ENGINEERING DRAFT DECEMBER 3, 2014 XXVI. LECTURE #26 Questions on homework and final Matlab exercise System Design Specification Approximation Implementation Filter Design 1) Analog filter (polynomial) mapping Impulse Invariant h[n]=h(nt) Run demobtwf.m in /mfls Bilinear (no aliasing) s = 1 T log z; z = est (10) s = 2 z 1 T z + 1 ; Ω = 2 T tan ω (11) 2 Run demosysd.m in /mfls 2) Windows Rectangular Hamming: Run wintst.m & wintstm.m in /mfiles 3) Design via Sampling the Ideal Interpolation functions h[n] = idfth[k], H(ω) = FTh[n] Run demorlft.m /mfls XXVII. LECTURE #27 Questions about the Matlab exercise Zero Phase Bi-directional IIR Run bidirex.m in /mfls Run demozpbt.m in /mfls Run tstbtw.m in /mfls Optimal mini-max systems Chebyshev polynomials Equiripple approximation Lagrange interpolation Run bpfirpm.m in /mfls Run optflt.m /mfls Run idlpfsam.m /mfls Alternation Theorem If P (e jω ) is a linear combination of r cosine functions, then a necessary and sufficient condition that P (e jω ) be the unique, best weighted Chebyshev approximation to a continuous function ˆD(e jω ) on A, a subset of (0, π), is that the weighted error function E(e jω ) exhibit at least r + 1 extremal frequencies in A, i.e., there must be r + 1 points on ω i in A such that ω 1 < ω 2... < ω r+1 and for i = 1, 2,..., r and E(e jωi ) = E(e jωi ); E(e jωi ) = max ω A [E(ejωi )] (12) Design Algorithm 1) Specify the desired response D(e jω ), the weighting function W (e j ω), and the length N 2) Formulate the approximation ˆD(e jω ), Ŵ (e jω ), P (e jω ) 3) Solve the approximation problem 4) Calculate the system s unit-sample response h[n] Find the mini-max (L norm) solution for P (e jω ) (firpm) 1) Guess r+1 extremal frequencies 2) Calculate δ (peak error magnitude) on the extremal set 3) Interpolate through the r points to define P (e jω ) 4) Calculate the error function E(e jω ) and find the local maxima where E(e jω ) δ 5) Retain the r+1 largest extrema 6) If the extremal frequencies have changed, go to step 2 XXVIII. LECTURE #28 Timing Run runrmadip.m in /mfls Run demobtwf.m in /mfls Run demosysd.m in /mfls Run fir12vspm.m in /mfls Run *.m below in /mfiles Matlab exercise dft4mat.m to show dft as matrix multiply (roundoff) optflt.m to investigate minimax behavior tstremez.m to design a minimax filter demobcel.m to compare butter, cheby, ellip, and minimax Auto-Correlation and Power Spectral Density Go over homework problems 10.1, 10.22, and Run prob1016a.m /mfls Run zpex.m in /mfls Run bpfirpm.m in /mfls Administer Quiz 10 XXIX. FINAL Monday 12/15, Whitaker 216, 3:30-5:30PM FINAL: 4 Problems of equal weight Emphasis on material since the test Closed book, notes, aid-memoir Tables from the text, as posted on the class website, will be included with the final Final covers: Material discussed in class Reading assignments and homework (& related) problems from the text

EE 521: Instrumentation and Measurements

EE 521: Instrumentation and Measurements Aly El-Osery Electrical Engineering Department, New Mexico Tech Socorro, New Mexico, USA November 1, 2009 1 / 27 1 The z-transform 2 Linear Time-Invariant System 3 Filter Design IIR Filters FIR Filters